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In sensor networks, nodes may malfunction due to the hostile environment. Therefore, dealing with node failure is a very important research issue. In this paper, we study distributed cooperative failure detection techniques. In the proposed techniques, the nodes around a suspected node collaborate with each other to reach an agreement on whether the suspect is faulty or malicious. We first formalize the problem as how to construct a dominating tree to cover all the neighbors of the suspect and give the lower bound of the message complexity. Two tree-based propagation collection protocols are proposed to construct dominating trees and collect information via the tree structure. Instead of using the traditional flooding technique, we propose a coverage-based heuristic to improve the system performance. Theoretical analysis and simulation results show that the heuristic can help achieve a higher tree coverage with lower message complexity, lower delay and lower energy consumption.